Buckling of Superconducting Structures: A Variational Approach Using the Law of Biot and Savart

Abstract

A structure of superconducting coils can collapse due to the Lorentz forces acting between the members of the structure, whenever the electric current through the structure exceeds a certain critical value, the so called buckling current. A method is presented based upon a variational principle, which uses as admissible fields those derived from the Biot-Savart law. This method combines the mathematical exactness of the variational principle with the straightforward availability of the Biot-Savart fields. Applications are presented for sets of n parallel rods (n ≥ 2) and for (finite or infinite) helical and spiral coils. For all these cases the buckling current is calculated and, moreover, some information about the buckling modes is provided. These buckling currents and modes are most easily found by using sinusoidal series representations for the buckling displacements. In all applications it is assumed that we deal with slender systems; the precise criterion for this is presented for each specific system. It turns out that for all systems considered in this paper the formula for the buckling current is globally the same; only a pre-factor differs in each case.